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In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the Geometric Satake Correspondence (due to Mirkovic-Vilonen, followed up in works by Anderson, Kamnitzer and many others). Here, one studies certain sheaves on the Affine Grassmannian for the group G and from this geometrical setup, one gets a canonical basis for a finite dimensional irrep of the Langlands dual group $G^\vee$.

Now, to someone who has studied the theory of finite dimensional representations of compact Lie Groups (Ex : can do tensor products, find branching laws etc, maybe not by the most efficient means, but by some means), is there any striking property of this theory that one can take as a hint of existence of the kind of canonical basis recalled above ?

Now, let me note here that if one takes the existence of the canonical basis, then one can prove non-trivial facts about finite dimensional representation theory in a nice way. For example Anderson (in this paper) proves a relationship between weight multiplicities and tensor product multiplicities using the MV basis. But, I don't think one can turn this argument in the other direction. That is, the existence of a relation between weight multiplicites and tensor product multiplicities is, by itself, not enough to indicate the existence of a canonical bases. I am looking for something in this direction.

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  • $\begingroup$ In many cases there are categorization results that show that canonical and dual canonical bases correspond to the basis of PIMs and simple modules in the (projective) Grothendieck groups for some (graded) algebra. This gives a direct representation theoretic construction of the canonical bases but it is not immediate in that proving this is hard, it doesn't work in all cases and even when it does work one has to use the "right" algebra to get the canonical bases (e.g. Hecke algebras rather than group algebras of symmetric groups). $\endgroup$ – Andrew May 27 '16 at 1:09
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I don't really see how to get there from just compact groups, so in that sense this is not an answer. My take on the question is something like: how might one have guessed the existence of canonical basis theories*, knowing only the Lie theory of, say, 1975? (The type A case was well under way by then, motivating standard monomial theory.)

The standard way to make the canonical basis uses quantum groups, and uses $q$ in a fundamental way. The closest we can get to a quantum group in the classical theory is the semiclassical limit, the Poisson structure on the Lie group. (Still too recent.) One question that becomes meaningful now is to ask what subgroups are Poisson subgroups. There are very few: I'll focus on the Borel $B$ (in particular, this is about complex groups not compact), whose importance was well-appreciated by then.

Now we can go beyond representation theory of $G$, i.e. complete linear systems over $G/B$, to talk about Demazure modules, complete linear systems over Schubert varieties $\overline{BwB}/B$, and look at the surjections $H^0(G/B; \mathcal L) \twoheadrightarrow H^0(\overline{BwB}/B;\mathcal L)$. In particular, one can ask for a basis for the irrep such that each of these kernels is spanned by a subset of the basis. Moreover, you can filter your basis of the kernel according to the order of vanishing along Schubert varieties, which suggests a means of indexing the basis. This line of thinking leads to the Lakshmibai-Seshadri conjecture (about characters, but motivated by the putative existence of such a basis).

*There's a problem with this question in that there are several bases, at the very least the canonical basis and the (different) semicanonical basis, the latter of which may or may not be the MV basis you asked about. Most of the striking properties you ask about hold for all of these. As such it's hard to find overlooked smoking guns that would pick out e.g. the MV basis over the others.

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  • $\begingroup$ Thinking of complex groups is perfectly fine. The spirit of my question was indeed along the lines of "what could have been said from classical results.. ". On the relation between Standard Monomial Theory and Canonical Bases, is such a relation true only for "the" canonical basis (coming from quantum groups) or does such a relation hold separately for each (semi?)-canonical basis. I am also unclear about the canonical vs semi-canonical distinction. Could you clarify that ? (by way of references or an edit to the answer) $\endgroup$ – Aswin May 27 '16 at 2:59
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    $\begingroup$ I'm pretty sure that the Demazure restriction result I'm talking about is for all three (two?) bases. The expansion of the canonical basis in semicanonical is positive upper-triangular. The canonical is defined more representation theoretically => using noncommutative algebras ~ using $\mathcal D$-modules => using the singular support of $\mathcal D$-modules, which are unions of conormal varieties. The semicanonical is defined using conormal varieties themselves. Hence the upper triangularity. $\endgroup$ – Allen Knutson May 27 '16 at 11:14

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